Calculation:

z = ((-5i)^(1/8))*(8^(1/3))

Result:



There are 24 solutions, due to “The Fundamental Theorem of Algebra”. Your expression contains square roots or powers to 1/n.

z1 = ((-5i)^(1/8))*(8^(1/3)) = 2.39869586-0.47713027i = 2.44568909 × ei (-0.0625) Calculation steps

  1. Divide: 1 / 8 = 0.125
  2. Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.22284455 × ei (-0.0625) = 1.19934793-0.23856514i
  3. Divide: 1 / 3 = 0.33333333
  4. Cube root: ∛8 = 2
  5. Multiple: the result of step No. 2 * the result of step No. 4 = (1.19934793-0.23856514i) * 2 = 2.39869586-0.47713027i
The result z1
Rectangular form:
z = 2.39869586-0.47713027i

Angle notation (phasor):
z = 2.44568909 ∠ -11°15'

Polar form:
z = 2.44568909 × (cos (-11°15') + i sin (-11°15'))

Exponential form:
z = 2.44568909 × ei (-0.0625)

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = -11.25° = -11°15' = -0.0625π ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = 2.39869586-0.47713027i
Real part: x = Re z = 2.399
Imaginary part: y = Im z = -0.47713027

z2 = ((-5i)^(1/8))*(8^(1/3)) = 2.03351616+1.35875206i = 2.44568909 × ei 3π/16 Calculation steps

  1. Divide: 1 / 8 = 0.125
  2. Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.22284455 × ei 3π/16 = 1.01675808+0.67937603i
  3. Divide: 1 / 3 = 0.33333333
  4. Cube root: ∛8 = 2
  5. Multiple: the result of step No. 2 * the result of step No. 4 = (1.01675808+0.67937603i) * 2 = 2.03351616+1.35875206i
The result z2
Rectangular form:
z = 2.03351616+1.35875206i

Angle notation (phasor):
z = 2.44568909 ∠ 33°45'

Polar form:
z = 2.44568909 × (cos 33°45' + i sin 33°45')

Exponential form:
z = 2.44568909 × ei 0.1875 = 2.44568909 × ei 3π/16

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = 33.75° = 33°45' = 0.1875π = 3π/16 ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = 2.03351616+1.35875206i
Real part: x = Re z = 2.034
Imaginary part: y = Im z = 1.35875206

z3 = ((-5i)^(1/8))*(8^(1/3)) = 0.47713027+2.39869586i = 2.44568909 × ei 7π/16 Calculation steps

  1. Divide: 1 / 8 = 0.125
  2. Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.22284455 × ei 7π/16 = 0.23856514+1.19934793i
  3. Divide: 1 / 3 = 0.33333333
  4. Cube root: ∛8 = 2
  5. Multiple: the result of step No. 2 * the result of step No. 4 = (0.23856514+1.19934793i) * 2 = 0.47713027+2.39869586i
The result z3
Rectangular form:
z = 0.47713027+2.39869586i

Angle notation (phasor):
z = 2.44568909 ∠ 78°45'

Polar form:
z = 2.44568909 × (cos 78°45' + i sin 78°45')

Exponential form:
z = 2.44568909 × ei 0.4375 = 2.44568909 × ei 7π/16

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = 78.75° = 78°45' = 0.4375π = 7π/16 ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = 0.47713027+2.39869586i
Real part: x = Re z = 0.477
Imaginary part: y = Im z = 2.39869586

z4 = ((-5i)^(1/8))*(8^(1/3)) = -1.35875206+2.03351616i = 2.44568909 × ei 11π/16 Calculation steps

  1. Divide: 1 / 8 = 0.125
  2. Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.22284455 × ei 11π/16 = -0.67937603+1.01675808i
  3. Divide: 1 / 3 = 0.33333333
  4. Cube root: ∛8 = 2
  5. Multiple: the result of step No. 2 * the result of step No. 4 = (-0.67937603+1.01675808i) * 2 = -1.35875206+2.03351616i
The result z4
Rectangular form:
z = -1.35875206+2.03351616i

Angle notation (phasor):
z = 2.44568909 ∠ 123°45'

Polar form:
z = 2.44568909 × (cos 123°45' + i sin 123°45')

Exponential form:
z = 2.44568909 × ei 0.6875 = 2.44568909 × ei 11π/16

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = 123.75° = 123°45' = 0.6875π = 11π/16 ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = -1.35875206+2.03351616i
Real part: x = Re z = -1.359
Imaginary part: y = Im z = 2.03351616

z5 = ((-5i)^(1/8))*(8^(1/3)) = -2.39869586+0.47713027i = 2.44568909 × ei 15π/16 Calculation steps

  1. Divide: 1 / 8 = 0.125
  2. Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.22284455 × ei 15π/16 = -1.19934793+0.23856514i
  3. Divide: 1 / 3 = 0.33333333
  4. Cube root: ∛8 = 2
  5. Multiple: the result of step No. 2 * the result of step No. 4 = (-1.19934793+0.23856514i) * 2 = -2.39869586+0.47713027i
The result z5
Rectangular form:
z = -2.39869586+0.47713027i

Angle notation (phasor):
z = 2.44568909 ∠ 168°45'

Polar form:
z = 2.44568909 × (cos 168°45' + i sin 168°45')

Exponential form:
z = 2.44568909 × ei 0.9375 = 2.44568909 × ei 15π/16

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = 168.75° = 168°45' = 0.9375π = 15π/16 ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = -2.39869586+0.47713027i
Real part: x = Re z = -2.399
Imaginary part: y = Im z = 0.47713027

z6 = ((-5i)^(1/8))*(8^(1/3)) = -2.03351616-1.35875206i = 2.44568909 × ei (-13π/16) Calculation steps

  1. Divide: 1 / 8 = 0.125
  2. Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.22284455 × ei (-13π/16) = -1.01675808-0.67937603i
  3. Divide: 1 / 3 = 0.33333333
  4. Cube root: ∛8 = 2
  5. Multiple: the result of step No. 2 * the result of step No. 4 = (-1.01675808-0.67937603i) * 2 = -2.03351616-1.35875206i
The result z6
Rectangular form:
z = -2.03351616-1.35875206i

Angle notation (phasor):
z = 2.44568909 ∠ -146°15'

Polar form:
z = 2.44568909 × (cos (-146°15') + i sin (-146°15'))

Exponential form:
z = 2.44568909 × ei (-0.8125) = 2.44568909 × ei (-13π/16)

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = -146.25° = -146°15' = -0.8125π = -13π/16 ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = -2.03351616-1.35875206i
Real part: x = Re z = -2.034
Imaginary part: y = Im z = -1.35875206

z7 = ((-5i)^(1/8))*(8^(1/3)) = -0.47713027-2.39869586i = 2.44568909 × ei (-9π/16) Calculation steps

  1. Divide: 1 / 8 = 0.125
  2. Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.22284455 × ei (-9π/16) = -0.23856514-1.19934793i
  3. Divide: 1 / 3 = 0.33333333
  4. Cube root: ∛8 = 2
  5. Multiple: the result of step No. 2 * the result of step No. 4 = (-0.23856514-1.19934793i) * 2 = -0.47713027-2.39869586i
The result z7
Rectangular form:
z = -0.47713027-2.39869586i

Angle notation (phasor):
z = 2.44568909 ∠ -101°15'

Polar form:
z = 2.44568909 × (cos (-101°15') + i sin (-101°15'))

Exponential form:
z = 2.44568909 × ei (-0.5625) = 2.44568909 × ei (-9π/16)

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = -101.25° = -101°15' = -0.5625π = -9π/16 ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = -0.47713027-2.39869586i
Real part: x = Re z = -0.477
Imaginary part: y = Im z = -2.39869586

z8 = ((-5i)^(1/8))*(8^(1/3)) = 1.35875206-2.03351616i = 2.44568909 × ei (-5π/16) Calculation steps

  1. Divide: 1 / 8 = 0.125
  2. Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.22284455 × ei (-5π/16) = 0.67937603-1.01675808i
  3. Divide: 1 / 3 = 0.33333333
  4. Cube root: ∛8 = 2
  5. Multiple: the result of step No. 2 * the result of step No. 4 = (0.67937603-1.01675808i) * 2 = 1.35875206-2.03351616i
The result z8
Rectangular form:
z = 1.35875206-2.03351616i

Angle notation (phasor):
z = 2.44568909 ∠ -56°15'

Polar form:
z = 2.44568909 × (cos (-56°15') + i sin (-56°15'))

Exponential form:
z = 2.44568909 × ei (-0.3125) = 2.44568909 × ei (-5π/16)

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = -56.25° = -56°15' = -0.3125π = -5π/16 ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = 1.35875206-2.03351616i
Real part: x = Re z = 1.359
Imaginary part: y = Im z = -2.03351616

z9 = ((-5i)^(1/8))*(8^(1/3)) = -0.78614099+2.31589669i = 2.44568909 × ei 29π/48 Calculation steps

  1. Divide: 1 / 8 = 0.125
  2. Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.22284455 × ei (-0.0625) = 1.19934793-0.23856514i
  3. Divide: 1 / 3 = 0.33333333
  4. Cube root: ∛8 = -1+1.73205081i
  5. Multiple: the result of step No. 2 * the result of step No. 4 = (1.19934793-0.23856514i) * (-1+1.73205081i) = 1.19934793 * (-1) + 1.19934793 * 1.7320508076i + (-0.2385651361i) * (-1) + (-0.2385651361i) * 1.7320508076i = -1.19934793+2.07733155i+0.23856514i-0.41320694i2 = -1.19934793+2.07733155i+0.23856514i+0.41320694 = -1.19934793 + 0.41320694 +i(2.07733155 + 0.23856514) = -0.78614099+2.31589669i
    alternative steps
    1.22284455 × ei (-0.0625) × 2 × ei 2π/3 = 1.2228445450497 × 2 × ei ((-0.0625)+2π/3) = 2.44568909 × ei 29π/48 = -0.78614099+2.31589669i
The result z9
Rectangular form:
z = -0.78614099+2.31589669i

Angle notation (phasor):
z = 2.44568909 ∠ 108°45'

Polar form:
z = 2.44568909 × (cos 108°45' + i sin 108°45')

Exponential form:
z = 2.44568909 × ei 0.6041667 = 2.44568909 × ei 29π/48

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = 108.75° = 108°45' = 0.6041667π = 29π/48 ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = -0.78614099+2.31589669i
Real part: x = Re z = -0.786
Imaginary part: y = Im z = 2.31589669

z10 = ((-5i)^(1/8))*(8^(1/3)) = -2.19347188+1.08170062i = 2.44568909 × ei 41π/48 Calculation steps

  1. Divide: 1 / 8 = 0.125
  2. Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.22284455 × ei 3π/16 = 1.01675808+0.67937603i
  3. Divide: 1 / 3 = 0.33333333
  4. Cube root: ∛8 = -1+1.73205081i
  5. Multiple: the result of step No. 2 * the result of step No. 4 = (1.01675808+0.67937603i) * (-1+1.73205081i) = 1.0167580798154 * (-1) + 1.0167580798154 * 1.7320508076i + 0.67937602882697i * (-1) + 0.67937602882697i * 1.7320508076i = -1.01675808+1.76107665i-0.67937603i+1.1767138i2 = -1.01675808+1.76107665i-0.67937603i-1.1767138 = -1.01675808 - 1.1767138 +i(1.76107665 - 0.67937603) = -2.19347188+1.08170062i
    alternative steps
    1.22284455 × ei 3π/16 × 2 × ei 2π/3 = 1.2228445450729 × 2 × ei (3π/16+2π/3) = 2.44568909 × ei 41π/48 = -2.19347188+1.08170062i
The result z10
Rectangular form:
z = -2.19347188+1.08170062i

Angle notation (phasor):
z = 2.44568909 ∠ 153°45'

Polar form:
z = 2.44568909 × (cos 153°45' + i sin 153°45')

Exponential form:
z = 2.44568909 × ei 0.8541667 = 2.44568909 × ei 41π/48

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = 153.75° = 153°45' = 0.8541667π = 41π/48 ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = -2.19347188+1.08170062i
Real part: x = Re z = -2.193
Imaginary part: y = Im z = 1.08170062

z11 = ((-5i)^(1/8))*(8^(1/3)) = -2.31589669-0.78614099i = 2.44568909 × ei (-43π/48) Calculation steps

  1. Divide: 1 / 8 = 0.125
  2. Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.22284455 × ei 7π/16 = 0.23856514+1.19934793i
  3. Divide: 1 / 3 = 0.33333333
  4. Cube root: ∛8 = -1+1.73205081i
  5. Multiple: the result of step No. 2 * the result of step No. 4 = (0.23856514+1.19934793i) * (-1+1.73205081i) = 0.2385651361 * (-1) + 0.2385651361 * 1.7320508076i + 1.19934793i * (-1) + 1.19934793i * 1.7320508076i = -0.23856514+0.41320694i-1.19934793i+2.07733155i2 = -0.23856514+0.41320694i-1.19934793i-2.07733155 = -0.23856514 - 2.07733155 +i(0.41320694 - 1.19934793) = -2.31589669-0.78614099i
    alternative steps
    1.22284455 × ei 7π/16 × 2 × ei 2π/3 = 1.2228445450497 × 2 × ei (7π/16+2π/3) = 2.44568909 × ei (-43π/48) = -2.31589669-0.78614099i
The result z11
Rectangular form:
z = -2.31589669-0.78614099i

Angle notation (phasor):
z = 2.44568909 ∠ -161°15'

Polar form:
z = 2.44568909 × (cos (-161°15') + i sin (-161°15'))

Exponential form:
z = 2.44568909 × ei (-0.8958333) = 2.44568909 × ei (-43π/48)

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = -161.25° = -161°15' = -0.8958333π = -43π/48 ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = -2.31589669-0.78614099i
Real part: x = Re z = -2.316
Imaginary part: y = Im z = -0.78614099

z12 = ((-5i)^(1/8))*(8^(1/3)) = -1.08170062-2.19347188i = 2.44568909 × ei (-31π/48) Calculation steps

  1. Divide: 1 / 8 = 0.125
  2. Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.22284455 × ei 11π/16 = -0.67937603+1.01675808i
  3. Divide: 1 / 3 = 0.33333333
  4. Cube root: ∛8 = -1+1.73205081i
  5. Multiple: the result of step No. 2 * the result of step No. 4 = (-0.67937603+1.01675808i) * (-1+1.73205081i) = -0.67937602882697 * (-1) + (-0.67937602882697) * 1.7320508076i + 1.0167580798154i * (-1) + 1.0167580798154i * 1.7320508076i = 0.67937603-1.1767138i-1.01675808i+1.76107665i2 = 0.67937603-1.1767138i-1.01675808i-1.76107665 = 0.67937603 - 1.76107665 +i(-1.1767138 - 1.01675808) = -1.08170062-2.19347188i
    alternative steps
    1.22284455 × ei 11π/16 × 2 × ei 2π/3 = 1.2228445450729 × 2 × ei (11π/16+2π/3) = 2.44568909 × ei (-31π/48) = -1.08170062-2.19347188i
The result z12
Rectangular form:
z = -1.08170062-2.19347188i

Angle notation (phasor):
z = 2.44568909 ∠ -116°15'

Polar form:
z = 2.44568909 × (cos (-116°15') + i sin (-116°15'))

Exponential form:
z = 2.44568909 × ei (-0.6458333) = 2.44568909 × ei (-31π/48)

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = -116.25° = -116°15' = -0.6458333π = -31π/48 ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = -1.08170062-2.19347188i
Real part: x = Re z = -1.082
Imaginary part: y = Im z = -2.19347188

z13 = ((-5i)^(1/8))*(8^(1/3)) = 0.78614099-2.31589669i = 2.44568909 × ei (-19π/48) Calculation steps

  1. Divide: 1 / 8 = 0.125
  2. Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.22284455 × ei 15π/16 = -1.19934793+0.23856514i
  3. Divide: 1 / 3 = 0.33333333
  4. Cube root: ∛8 = -1+1.73205081i
  5. Multiple: the result of step No. 2 * the result of step No. 4 = (-1.19934793+0.23856514i) * (-1+1.73205081i) = -1.19934793 * (-1) + (-1.19934793) * 1.7320508076i + 0.2385651361i * (-1) + 0.2385651361i * 1.7320508076i = 1.19934793-2.07733155i-0.23856514i+0.41320694i2 = 1.19934793-2.07733155i-0.23856514i-0.41320694 = 1.19934793 - 0.41320694 +i(-2.07733155 - 0.23856514) = 0.78614099-2.31589669i
    alternative steps
    1.22284455 × ei 15π/16 × 2 × ei 2π/3 = 1.2228445450497 × 2 × ei (15π/16+2π/3) = 2.44568909 × ei (-19π/48) = 0.78614099-2.31589669i
The result z13
Rectangular form:
z = 0.78614099-2.31589669i

Angle notation (phasor):
z = 2.44568909 ∠ -71°15'

Polar form:
z = 2.44568909 × (cos (-71°15') + i sin (-71°15'))

Exponential form:
z = 2.44568909 × ei (-0.3958333) = 2.44568909 × ei (-19π/48)

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = -71.25° = -71°15' = -0.3958333π = -19π/48 ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = 0.78614099-2.31589669i
Real part: x = Re z = 0.786
Imaginary part: y = Im z = -2.31589669

z14 = ((-5i)^(1/8))*(8^(1/3)) = 2.19347188-1.08170062i = 2.44568909 × ei (-7π/48) Calculation steps

  1. Divide: 1 / 8 = 0.125
  2. Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.22284455 × ei (-13π/16) = -1.01675808-0.67937603i
  3. Divide: 1 / 3 = 0.33333333
  4. Cube root: ∛8 = -1+1.73205081i
  5. Multiple: the result of step No. 2 * the result of step No. 4 = (-1.01675808-0.67937603i) * (-1+1.73205081i) = -1.0167580798154 * (-1) + (-1.0167580798154) * 1.7320508076i + (-0.67937602882697i) * (-1) + (-0.67937602882697i) * 1.7320508076i = 1.01675808-1.76107665i+0.67937603i-1.1767138i2 = 1.01675808-1.76107665i+0.67937603i+1.1767138 = 1.01675808 + 1.1767138 +i(-1.76107665 + 0.67937603) = 2.19347188-1.08170062i
    alternative steps
    1.22284455 × ei (-13π/16) × 2 × ei 2π/3 = 1.2228445450729 × 2 × ei ((-13π/16)+2π/3) = 2.44568909 × ei (-7π/48) = 2.19347188-1.08170062i
The result z14
Rectangular form:
z = 2.19347188-1.08170062i

Angle notation (phasor):
z = 2.44568909 ∠ -26°15'

Polar form:
z = 2.44568909 × (cos (-26°15') + i sin (-26°15'))

Exponential form:
z = 2.44568909 × ei (-0.1458333) = 2.44568909 × ei (-7π/48)

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = -26.25° = -26°15' = -0.1458333π = -7π/48 ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = 2.19347188-1.08170062i
Real part: x = Re z = 2.193
Imaginary part: y = Im z = -1.08170062

z15 = ((-5i)^(1/8))*(8^(1/3)) = 2.31589669+0.78614099i = 2.44568909 × ei 0.1041667 Calculation steps

  1. Divide: 1 / 8 = 0.125
  2. Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.22284455 × ei (-9π/16) = -0.23856514-1.19934793i
  3. Divide: 1 / 3 = 0.33333333
  4. Cube root: ∛8 = -1+1.73205081i
  5. Multiple: the result of step No. 2 * the result of step No. 4 = (-0.23856514-1.19934793i) * (-1+1.73205081i) = -0.2385651361 * (-1) + (-0.2385651361) * 1.7320508076i + (-1.19934793i) * (-1) + (-1.19934793i) * 1.7320508076i = 0.23856514-0.41320694i+1.19934793i-2.07733155i2 = 0.23856514-0.41320694i+1.19934793i+2.07733155 = 0.23856514 + 2.07733155 +i(-0.41320694 + 1.19934793) = 2.31589669+0.78614099i
    alternative steps
    1.22284455 × ei (-9π/16) × 2 × ei 2π/3 = 1.2228445450497 × 2 × ei ((-9π/16)+2π/3) = 2.44568909 × ei 0.1041667 = 2.31589669+0.78614099i
The result z15
Rectangular form:
z = 2.31589669+0.78614099i

Angle notation (phasor):
z = 2.44568909 ∠ 18°45'

Polar form:
z = 2.44568909 × (cos 18°45' + i sin 18°45')

Exponential form:
z = 2.44568909 × ei 0.1041667

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = 18.75° = 18°45' = 0.1041667π ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = 2.31589669+0.78614099i
Real part: x = Re z = 2.316
Imaginary part: y = Im z = 0.78614099

z16 = ((-5i)^(1/8))*(8^(1/3)) = 1.08170062+2.19347188i = 2.44568909 × ei 17π/48 Calculation steps

  1. Divide: 1 / 8 = 0.125
  2. Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.22284455 × ei (-5π/16) = 0.67937603-1.01675808i
  3. Divide: 1 / 3 = 0.33333333
  4. Cube root: ∛8 = -1+1.73205081i
  5. Multiple: the result of step No. 2 * the result of step No. 4 = (0.67937603-1.01675808i) * (-1+1.73205081i) = 0.67937602882697 * (-1) + 0.67937602882697 * 1.7320508076i + (-1.0167580798154i) * (-1) + (-1.0167580798154i) * 1.7320508076i = -0.67937603+1.1767138i+1.01675808i-1.76107665i2 = -0.67937603+1.1767138i+1.01675808i+1.76107665 = -0.67937603 + 1.76107665 +i(1.1767138 + 1.01675808) = 1.08170062+2.19347188i
    alternative steps
    1.22284455 × ei (-5π/16) × 2 × ei 2π/3 = 1.2228445450729 × 2 × ei ((-5π/16)+2π/3) = 2.44568909 × ei 17π/48 = 1.08170062+2.19347188i
The result z16
Rectangular form:
z = 1.08170062+2.19347188i

Angle notation (phasor):
z = 2.44568909 ∠ 63°45'

Polar form:
z = 2.44568909 × (cos 63°45' + i sin 63°45')

Exponential form:
z = 2.44568909 × ei 0.3541667 = 2.44568909 × ei 17π/48

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = 63.75° = 63°45' = 0.3541667π = 17π/48 ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = 1.08170062+2.19347188i
Real part: x = Re z = 1.082
Imaginary part: y = Im z = 2.19347188

z17 = ((-5i)^(1/8))*(8^(1/3)) = -1.61255487-1.83876641i = 2.44568909 × ei (-35π/48) Calculation steps

  1. Divide: 1 / 8 = 0.125
  2. Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.22284455 × ei (-0.0625) = 1.19934793-0.23856514i
  3. Divide: 1 / 3 = 0.33333333
  4. Cube root: ∛8 = -1-1.73205081i
  5. Multiple: the result of step No. 2 * the result of step No. 4 = (1.19934793-0.23856514i) * (-1-1.73205081i) = 1.19934793 * (-1) + 1.19934793 * (-1.7320508076i) + (-0.2385651361i) * (-1) + (-0.2385651361i) * (-1.7320508076i) = -1.19934793-2.07733155i+0.23856514i+0.41320694i2 = -1.19934793-2.07733155i+0.23856514i-0.41320694 = -1.19934793 - 0.41320694 +i(-2.07733155 + 0.23856514) = -1.61255487-1.83876641i
    alternative steps
    1.22284455 × ei (-0.0625) × 2 × ei (-2π/3) = 1.2228445450497 × 2 × ei ((-0.0625)+(-2π/3)) = 2.44568909 × ei (-35π/48) = -1.61255487-1.83876641i
The result z17
Rectangular form:
z = -1.61255487-1.83876641i

Angle notation (phasor):
z = 2.44568909 ∠ -131°15'

Polar form:
z = 2.44568909 × (cos (-131°15') + i sin (-131°15'))

Exponential form:
z = 2.44568909 × ei (-0.7291667) = 2.44568909 × ei (-35π/48)

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = -131.25° = -131°15' = -0.7291667π = -35π/48 ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = -1.61255487-1.83876641i
Real part: x = Re z = -1.613
Imaginary part: y = Im z = -1.83876641

z18 = ((-5i)^(1/8))*(8^(1/3)) = 0.15995572-2.44045268i = 2.44568909 × ei (-23π/48) Calculation steps

  1. Divide: 1 / 8 = 0.125
  2. Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.22284455 × ei 3π/16 = 1.01675808+0.67937603i
  3. Divide: 1 / 3 = 0.33333333
  4. Cube root: ∛8 = -1-1.73205081i
  5. Multiple: the result of step No. 2 * the result of step No. 4 = (1.01675808+0.67937603i) * (-1-1.73205081i) = 1.0167580798154 * (-1) + 1.0167580798154 * (-1.7320508076i) + 0.67937602882697i * (-1) + 0.67937602882697i * (-1.7320508076i) = -1.01675808-1.76107665i-0.67937603i-1.1767138i2 = -1.01675808-1.76107665i-0.67937603i+1.1767138 = -1.01675808 + 1.1767138 +i(-1.76107665 - 0.67937603) = 0.15995572-2.44045268i
    alternative steps
    1.22284455 × ei 3π/16 × 2 × ei (-2π/3) = 1.2228445450729 × 2 × ei (3π/16+(-2π/3)) = 2.44568909 × ei (-23π/48) = 0.15995572-2.44045268i
The result z18
Rectangular form:
z = 0.15995572-2.44045268i

Angle notation (phasor):
z = 2.44568909 ∠ -86°15'

Polar form:
z = 2.44568909 × (cos (-86°15') + i sin (-86°15'))

Exponential form:
z = 2.44568909 × ei (-0.4791667) = 2.44568909 × ei (-23π/48)

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = -86.25° = -86°15' = -0.4791667π = -23π/48 ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = 0.15995572-2.44045268i
Real part: x = Re z = 0.16
Imaginary part: y = Im z = -2.44045268

z19 = ((-5i)^(1/8))*(8^(1/3)) = 1.83876641-1.61255487i = 2.44568909 × ei (-11π/48) Calculation steps

  1. Divide: 1 / 8 = 0.125
  2. Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.22284455 × ei 7π/16 = 0.23856514+1.19934793i
  3. Divide: 1 / 3 = 0.33333333
  4. Cube root: ∛8 = -1-1.73205081i
  5. Multiple: the result of step No. 2 * the result of step No. 4 = (0.23856514+1.19934793i) * (-1-1.73205081i) = 0.2385651361 * (-1) + 0.2385651361 * (-1.7320508076i) + 1.19934793i * (-1) + 1.19934793i * (-1.7320508076i) = -0.23856514-0.41320694i-1.19934793i-2.07733155i2 = -0.23856514-0.41320694i-1.19934793i+2.07733155 = -0.23856514 + 2.07733155 +i(-0.41320694 - 1.19934793) = 1.83876641-1.61255487i
    alternative steps
    1.22284455 × ei 7π/16 × 2 × ei (-2π/3) = 1.2228445450497 × 2 × ei (7π/16+(-2π/3)) = 2.44568909 × ei (-11π/48) = 1.83876641-1.61255487i
The result z19
Rectangular form:
z = 1.83876641-1.61255487i

Angle notation (phasor):
z = 2.44568909 ∠ -41°15'

Polar form:
z = 2.44568909 × (cos (-41°15') + i sin (-41°15'))

Exponential form:
z = 2.44568909 × ei (-0.2291667) = 2.44568909 × ei (-11π/48)

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = -41.25° = -41°15' = -0.2291667π = -11π/48 ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = 1.83876641-1.61255487i
Real part: x = Re z = 1.839
Imaginary part: y = Im z = -1.61255487

z20 = ((-5i)^(1/8))*(8^(1/3)) = 2.44045268+0.15995572i = 2.44568909 × ei 0.0208333 Calculation steps

  1. Divide: 1 / 8 = 0.125
  2. Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.22284455 × ei 11π/16 = -0.67937603+1.01675808i
  3. Divide: 1 / 3 = 0.33333333
  4. Cube root: ∛8 = -1-1.73205081i
  5. Multiple: the result of step No. 2 * the result of step No. 4 = (-0.67937603+1.01675808i) * (-1-1.73205081i) = -0.67937602882697 * (-1) + (-0.67937602882697) * (-1.7320508076i) + 1.0167580798154i * (-1) + 1.0167580798154i * (-1.7320508076i) = 0.67937603+1.1767138i-1.01675808i-1.76107665i2 = 0.67937603+1.1767138i-1.01675808i+1.76107665 = 0.67937603 + 1.76107665 +i(1.1767138 - 1.01675808) = 2.44045268+0.15995572i
    alternative steps
    1.22284455 × ei 11π/16 × 2 × ei (-2π/3) = 1.2228445450729 × 2 × ei (11π/16+(-2π/3)) = 2.44568909 × ei 0.0208333 = 2.44045268+0.15995572i
The result z20
Rectangular form:
z = 2.44045268+0.15995572i

Angle notation (phasor):
z = 2.44568909 ∠ 3°45'

Polar form:
z = 2.44568909 × (cos 3°45' + i sin 3°45')

Exponential form:
z = 2.44568909 × ei 0.0208333

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = 3.75° = 3°45' = 0.0208333π ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = 2.44045268+0.15995572i
Real part: x = Re z = 2.44
Imaginary part: y = Im z = 0.15995572

z21 = ((-5i)^(1/8))*(8^(1/3)) = 1.61255487+1.83876641i = 2.44568909 × ei 13π/48 Calculation steps

  1. Divide: 1 / 8 = 0.125
  2. Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.22284455 × ei 15π/16 = -1.19934793+0.23856514i
  3. Divide: 1 / 3 = 0.33333333
  4. Cube root: ∛8 = -1-1.73205081i
  5. Multiple: the result of step No. 2 * the result of step No. 4 = (-1.19934793+0.23856514i) * (-1-1.73205081i) = -1.19934793 * (-1) + (-1.19934793) * (-1.7320508076i) + 0.2385651361i * (-1) + 0.2385651361i * (-1.7320508076i) = 1.19934793+2.07733155i-0.23856514i-0.41320694i2 = 1.19934793+2.07733155i-0.23856514i+0.41320694 = 1.19934793 + 0.41320694 +i(2.07733155 - 0.23856514) = 1.61255487+1.83876641i
    alternative steps
    1.22284455 × ei 15π/16 × 2 × ei (-2π/3) = 1.2228445450497 × 2 × ei (15π/16+(-2π/3)) = 2.44568909 × ei 13π/48 = 1.61255487+1.83876641i
The result z21
Rectangular form:
z = 1.61255487+1.83876641i

Angle notation (phasor):
z = 2.44568909 ∠ 48°45'

Polar form:
z = 2.44568909 × (cos 48°45' + i sin 48°45')

Exponential form:
z = 2.44568909 × ei 0.2708333 = 2.44568909 × ei 13π/48

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = 48.75° = 48°45' = 0.2708333π = 13π/48 ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = 1.61255487+1.83876641i
Real part: x = Re z = 1.613
Imaginary part: y = Im z = 1.83876641

z22 = ((-5i)^(1/8))*(8^(1/3)) = -0.15995572+2.44045268i = 2.44568909 × ei 25π/48 Calculation steps

  1. Divide: 1 / 8 = 0.125
  2. Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.22284455 × ei (-13π/16) = -1.01675808-0.67937603i
  3. Divide: 1 / 3 = 0.33333333
  4. Cube root: ∛8 = -1-1.73205081i
  5. Multiple: the result of step No. 2 * the result of step No. 4 = (-1.01675808-0.67937603i) * (-1-1.73205081i) = -1.0167580798154 * (-1) + (-1.0167580798154) * (-1.7320508076i) + (-0.67937602882697i) * (-1) + (-0.67937602882697i) * (-1.7320508076i) = 1.01675808+1.76107665i+0.67937603i+1.1767138i2 = 1.01675808+1.76107665i+0.67937603i-1.1767138 = 1.01675808 - 1.1767138 +i(1.76107665 + 0.67937603) = -0.15995572+2.44045268i
    alternative steps
    1.22284455 × ei (-13π/16) × 2 × ei (-2π/3) = 1.2228445450729 × 2 × ei ((-13π/16)+(-2π/3)) = 2.44568909 × ei 25π/48 = -0.15995572+2.44045268i
The result z22
Rectangular form:
z = -0.15995572+2.44045268i

Angle notation (phasor):
z = 2.44568909 ∠ 93°45'

Polar form:
z = 2.44568909 × (cos 93°45' + i sin 93°45')

Exponential form:
z = 2.44568909 × ei 0.5208333 = 2.44568909 × ei 25π/48

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = 93.75° = 93°45' = 0.5208333π = 25π/48 ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = -0.15995572+2.44045268i
Real part: x = Re z = -0.16
Imaginary part: y = Im z = 2.44045268

z23 = ((-5i)^(1/8))*(8^(1/3)) = -1.83876641+1.61255487i = 2.44568909 × ei 37π/48 Calculation steps

  1. Divide: 1 / 8 = 0.125
  2. Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.22284455 × ei (-9π/16) = -0.23856514-1.19934793i
  3. Divide: 1 / 3 = 0.33333333
  4. Cube root: ∛8 = -1-1.73205081i
  5. Multiple: the result of step No. 2 * the result of step No. 4 = (-0.23856514-1.19934793i) * (-1-1.73205081i) = -0.2385651361 * (-1) + (-0.2385651361) * (-1.7320508076i) + (-1.19934793i) * (-1) + (-1.19934793i) * (-1.7320508076i) = 0.23856514+0.41320694i+1.19934793i+2.07733155i2 = 0.23856514+0.41320694i+1.19934793i-2.07733155 = 0.23856514 - 2.07733155 +i(0.41320694 + 1.19934793) = -1.83876641+1.61255487i
    alternative steps
    1.22284455 × ei (-9π/16) × 2 × ei (-2π/3) = 1.2228445450497 × 2 × ei ((-9π/16)+(-2π/3)) = 2.44568909 × ei 37π/48 = -1.83876641+1.61255487i
The result z23
Rectangular form:
z = -1.83876641+1.61255487i

Angle notation (phasor):
z = 2.44568909 ∠ 138°45'

Polar form:
z = 2.44568909 × (cos 138°45' + i sin 138°45')

Exponential form:
z = 2.44568909 × ei 0.7708333 = 2.44568909 × ei 37π/48

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = 138.75° = 138°45' = 0.7708333π = 37π/48 ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = -1.83876641+1.61255487i
Real part: x = Re z = -1.839
Imaginary part: y = Im z = 1.61255487

z24 = ((-5i)^(1/8))*(8^(1/3)) = -2.44045268-0.15995572i = 2.44568909 × ei (-47π/48) Calculation steps

  1. Divide: 1 / 8 = 0.125
  2. Exponentiation: (-5i) ^ the result of step No. 1 = (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.22284455 × ei (-5π/16) = 0.67937603-1.01675808i
  3. Divide: 1 / 3 = 0.33333333
  4. Cube root: ∛8 = -1-1.73205081i
  5. Multiple: the result of step No. 2 * the result of step No. 4 = (0.67937603-1.01675808i) * (-1-1.73205081i) = 0.67937602882697 * (-1) + 0.67937602882697 * (-1.7320508076i) + (-1.0167580798154i) * (-1) + (-1.0167580798154i) * (-1.7320508076i) = -0.67937603-1.1767138i+1.01675808i+1.76107665i2 = -0.67937603-1.1767138i+1.01675808i-1.76107665 = -0.67937603 - 1.76107665 +i(-1.1767138 + 1.01675808) = -2.44045268-0.15995572i
    alternative steps
    1.22284455 × ei (-5π/16) × 2 × ei (-2π/3) = 1.2228445450729 × 2 × ei ((-5π/16)+(-2π/3)) = 2.44568909 × ei (-47π/48) = -2.44045268-0.15995572i
The result z24
Rectangular form:
z = -2.44045268-0.15995572i

Angle notation (phasor):
z = 2.44568909 ∠ -176°15'

Polar form:
z = 2.44568909 × (cos (-176°15') + i sin (-176°15'))

Exponential form:
z = 2.44568909 × ei (-0.9791667) = 2.44568909 × ei (-47π/48)

Polar coordinates:
r = |z| = 2.4456891 ... magnitude (modulus, absolute value)
θ = arg z = -176.25° = -176°15' = -0.9791667π = -47π/48 ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = -2.44045268-0.15995572i
Real part: x = Re z = -2.44
Imaginary part: y = Im z = -0.15995572

Calculate next expression:






This calculator does basic arithmetic on complex numbers and evaluates expressions in the set of complex numbers. As imaginary unit use i or j (in electrical engineering), which satisfies basic equation i2 = −1 or j2 = −1. The calculator also provides conversion of a complex number into angle notation (phasor notation), exponential, or polar coordinates (magnitude and angle). Enter expression with complex numbers like 5*(1+i)(-2-5i)^2

Complex numbers in the angle notation or phasor (polar coordinates r, θ) may you write as rLθ where r is magnitude/amplitude/radius, and θ is angle (phase) in degrees, for example, 5L65 which is same as 5*cis(65°).
Example of multiplication of two imaginary numbers in the angle/polar/phasor notation: 10L45 * 3L90.

Why the next complex numbers calculator when we have WolframAlpha? Because Wolfram tool is slow and some features such as step by step are charged premium service.                  
For use in education (for example, calculations of alternating currents at high school), you need a quick and precise complex number calculator.



Basic operations with complex numbers

We hope that work with the complex number is quite easy because you can work with imaginary unit i as a variable. And use definition i2 = -1 to simplify complex expressions. Many operations are the same as operations with two-dimensional vectors.

Addition

Very simple, add up the real parts (without i) and add up the imaginary parts (with i):
This is equal to use rule: (a+bi)+(c+di) = (a+c) + (b+d)i

(1+i) + (6-5i) = 7-4i
12 + 6-5i = 18-5i
(10-5i) + (-5+5i) = 5

Subtraction

Again very simple, subtract the real parts and subtract the imaginary parts (with i):
This is equal to use rule: (a+bi)+(c+di) = (a-c) + (b-d)i

(1+i) - (3-5i) = -2+6i
-1/2 - (6-5i) = -6.5+5i
(10-5i) - (-5+5i) = 15-10i

Multiplication

To multiply two complex number use distributive law, avoid binomials and apply i2 = -1.
This is equal to use rule: (a+bi)(c+di) = (ac-bd) + (ad+bc)i

(1+i) (3+5i) = 1*3+1*5i+i*3+i*5i = 3+5i+3i-5 = -2+8i
-1/2 * (6-5i) = -3+2.5i
(10-5i) * (-5+5i) = -25+75i

Division

Division of two complex number is based on avoid imaginary unit i from denominator. This can be done only via i2 = -1. If denominator is c+di, to make it without i (or make it real), just multiply with conjugate c-di:

(c+di)(c-di) = c2+d2

a+bic+di=(a+bi)(cdi)(c+di)(cdi)=ac+bd+i(bcad)c2+d2=ac+bdc2+d2+bcadc2+d2i\dfrac{a+bi}{c+di} = \dfrac{(a+bi)(c-di)}{(c+di)(c-di)} = \dfrac{ac+bd+i(bc-ad)}{c^2+d^2} = \dfrac{ac+bd}{c^2+d^2}+\dfrac{bc-ad}{c^2+d^2} i

(10-5i) / (1+i) = 2.5-7.5i
-3 / (2-i) = -1.2-0.6i
6i / (4+3i) = 0.72+0.96i

Absolute value or modulus

Absolute value or modulus is distance of image of complex number from origin in plane. That use Pythagorean theorem, just as case of 2D vector. Very simple, see examples: |3+4i| = 5
|1-i| = 1.4142135623731
|6i| = 6
abs(2+5i) = 5.3851648071345

Square root

Square root of complex number (a+bi) is z, if z2 = (a+bi). Here ends simplicity. Because of the fundamental theorem of algebra, you will always have two different square roots for a given number. If you want to find out the possible values, the easiest way is probably to go with De Moivre's formula. Here our calculator is on edge, because square root is not a well defined function on complex number. We calculate all complex roots from any number - even in expressions:

sqrt(9i) = 2.12132034+2.12132034i
sqrt(10-6i) = 3.29104116-0.91156563i
pow(-32,1/5)/5 = -0.4
pow(1+2i,1/3)*sqrt(4) = 2.43923302+0.94342254i
pow(-5i,1/8)*pow(8,1/3) = 2.39869586-0.47713027i

Square, power, complex exponentiation

Yes, our calculator can power any complex number to any integer (positive, negative), real or even complex number. In another words, we calculate 'complex number to a complex power' or 'complex number raised to a power'...
Famous example:
ii=eπ/2i^i = e^{-\pi/2}
i^2 = -1
i^61 = i
(6-2i)^6 = -22528-59904i
(6-i)^4.5 = 2486.13779853-2284.55578905i
(6-5i)^(-3+32i) = 2929449.0670531-9022199.6661184i
i^i = 0.2078795764
pow(1+i,3) = -2+2i

Functions

sqrt
Square Root of a value or expression.
sin
sine of a value or expression. Autodetect radians/degrees.
cos
cosine of a value or expression. Autodetect radians/degrees.
tan/tg
tangent of a value or expression. Autodetect radians/degrees.
exp
e (the Euler Constant) raised to the power of a value or expression
pow
Power one complex number to another integer/real/comple number
ln
The natural logarithm of a value or expression
log
The base-10 logarithm of a value or expression
abs or |1+i|
Absolute value of a value or expression
phase
Phase (angle) of a complex number
cis
is less known notation: cis(x) = cos(x)+ i sin(x); example: cis (pi/2) + 3 = 3+i
conj
conjugate of complex number - example: conj(4i+5) = 5-4i

Examples:

cube root: cuberoot(1-27i)
roots of Complex Numbers: pow(1+i,1/7)
phase, complex number angle: phase(1+i)
cis form complex numbers: 5*cis(45°)
The polar form of complex numbers: 10L60
complex conjugate calculator: conj(4+5i)
equation with complex numbers: (z+i/2 )/(1-i) = 4z+5i
system of equations with imaginary numbers: x-y = 4+6i; 3ix+7y=x+iy
De Moivre's theorem - equation: z^4=1
multiplication of three complex numbers: (1+3i)(3+4i)(−5+3i)
Find the product of 3-4i and its conjugate.: (3-4i)*conj(3-4i)
operations with complex numbers: (3-i)^3